What is the integral of e^{-4x}sin(x) ?

The integral of e^{-4x}sin(x) is -(e^{-4x}cos(x))/(17)-(4e^{-4x}sin(x))/(17)+C

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integral of e^{-4x}sin(x)

\int e^{- 4 x} sin (x) d x

- \frac{e ^{- 4 x} cos ( x )}{17} - \frac{4 e ^{- 4 x} sin ( x )}{17} + C

Solution steps

\int e^{- 4 x} sin (x) d x

Apply Integration By Parts

: - e^{- 4 x} cos (x) - \int 4 e^{- 4 x} cos (x) d x

= - e^{- 4 x} cos (x) - \int 4 e^{- 4 x} cos (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - e^{- 4 x} cos (x) - 4 \cdot \int e^{- 4 x} cos (x) d x

Apply Integration By Parts

: e^{- 4 x} sin (x) - \int - 4 e^{- 4 x} sin (x) d x

= - e^{- 4 x} cos (x) - 4 (e^{- 4 x} sin (x) - \int - 4 e^{- 4 x} sin (x) d x)

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - e^{- 4 x} cos (x) - 4 (e^{- 4 x} sin (x) - (- 4 \cdot \int e^{- 4 x} sin (x) d x))

Therefore

\int e^{- 4 x} sin (x) d x = - e^{- 4 x} cos (x) - 4 (e^{- 4 x} sin (x) - (- 4 \cdot \int e^{- 4 x} sin (x) d x))

Isolate

\int e^{- 4 x} sin (x) d x

= - \frac{e ^{- 4 x} cos ( x )}{17} - \frac{4 e ^{- 4 x} sin ( x )}{17}

Add a constant to the solution

= - \frac{e ^{- 4 x} cos ( x )}{17} - \frac{4 e ^{- 4 x} sin ( x )}{17} + C

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What is the integral of e^{-4x}sin(x) ?
The integral of e^{-4x}sin(x) is -(e^{-4x}cos(x))/(17)-(4e^{-4x}sin(x))/(17)+C